Calculus:  A Modern Approach


Jeff Knisley and Kevin Shirley[1]




          Calculus occupies a pivotal position in the mathematics curriculum.  It is the mathematical foundation for much of the science, mathematics, and engineering curriculum at a university.  For the aspiring mathematics student, it is a first exposure to rigorous mathematics.  For the future engineer, it is an introduction to the modeling and approximation techniques used throughout an engineering curriculum.  For the future scientist, it is the mathematical language that will be used to express many of the most important scientific concepts. 

          Consequently, it is imperative that calculus be presented as it is used and understood by today’s engineers, scientists, and mathematicians.  Rigor in calculus should prepare students for rigor in higher-level mathematics courses.  Modeling and approximation in calculus should resemble the techniques and methods currently in use.  Concepts, definitions, terminology, and interpretation in calculus should be as current as possible whenever possible. 

          Although a completely modern calculus text is neither possible nor perhaps desirable, our motivation for writing this textbook was to present calculus as the foundation of modern science, engineering, and mathematics.  To accomplish such a goal, however, we realized that such a textbook would also need to address pedagogical issues, as well as issues related to the use of technology, the rule of three, and other similar issues. 

          In particular, we began this textbook by identifying 3 issues central to the development of a more modern and truly effective calculus text:

1.        How students learn mathematics, and in particular, calculus.

2.        How calculus is used and conceptualized in modern science, engineering, and mathematics 

3.        What combination of technology, reform methods, and traditional techniques best address 1 and 2

In addition, we wanted to create a textbook that had a coherent structure that allowed ideas to flow from one section to the next. 

          Once we had addressed these issues to our satisfaction, we developed a comprehensive plan for producing the best possible textbook.  The plan we developed is almost a book itself, and parts of the plan have been published or submitted for publication in scholarly journals.  The original versions of the plan documents can be found at .  In addition to the plan, we also redeveloped many calculus concepts to reflect modern thinking about those concepts, and we also developed a list of all the topics that we felt should be included in a calculus course that reflects the needs of today’s mathematicians and scientists.

          Finally, we used these materials to write the actual textbook.  It combines technology, reform, and tradition in a way that we feel best serves today’s students.  It is based on research into how students learn mathematics.  Most importantly, it uses relevant applications and reformulated definitions to present calculus as the foundation of modern mathematics, science and engineering.


Incorporating Research into Mathematical Learning

          Soon after we began exploring how students learn mathematics and calculus, we realized that the first few chapters would have to differ markedly from traditional and even reformed approaches.  For example, several studies have shown that even our best calculus students fail to grasp the limit concept (several such studies have appeared over the past decade in the Journal for Research in Mathematics Education).  Many of the unique features in the first 2 were designed to address these shortcomings in learning limits.

          Studies have also shown that although each person has their own unique learning style, there are some aspects of learning mathematics that all of us have in common.  Based on these commonalities, we developed a model of how students learn mathematics.  Details of this model can be found at at To summarize, our efforts lead to the following 4 principles:

·Concepts should be introduced in as simple a setting as possible

·Definitions should be developed and utilized as soon as possible

·Concepts should be reinforced with recurring themes, written assignments, and technology

·Computation and rigor are important goals in the learning of mathematics

A framework for the textbook was then constructed and reconstructed until we felt that it best reflected these 4 principles.

          For example, the first principle—that concepts should be introduced in as simple a setting as possible—led us to introduce limits, derivatives, tangent lines, and rates of change in the simple setting of polynomials.  This approach allows us to establish fundamental ideas in calculus very early.  Students are using derivative rules by the second week of the course, and they have been introduced to the recurring themes of the limits, the chain rule, and differential equations by the end of the third week. 

          However, we do not cater to students, nor do we compromise the presentation of calculus in any way.  Once students have been exposed to differential calculus in the context of polynomials, chapter two presents them with rigorous definitions and proofs of basic theorems.  By chapter 3, Applications of the Derivative, students have encountered all of the material they would have encountered in a traditional approach—and then some—but without much of the confusion and frustration they might have otherwise developed.

          Perhaps as importantly, this approach allows us to revisit the fundamental concepts of calculus over and over again.  For example, the chain rule is explicitly revisited at least twice in each of the first, second, third, fourth, sixth, and ninth chapters.  Monotonicity and concavity are revisited in three different chapters.  Limits are reviewed and revisited time and time again, and the definition of the integral occurs in a vast array of settings.

          We call such repetition the use of recurring themes, and we call the use of recurring themes a spiraling approach to calculus.  After having used this textbook for the past 4 years, we have found that the average student tends to master the derivative rules, including the chain rule.  They also tend to have a rather sophisticated understanding of the many different roles of the derivative, and they have at least a working knowledge of what limits are all about.  

          In addition, many of the components of the book are designed to reduce the frustration and confusion expressed by so many students when trying to learn mathematics.  Many of the examples were developed in concert with the exercise sets, and many of the sections were developed to not only introduce new ideas, but also to reinforce ideas from earlier sections.   

          Finally, spiraling and the use of recurring themes make the book very flexible.  Some sections can be covered less rigorously than others, because many of the ideas presented in a given section will occur again in a later section.  The organization of the text further enhances this flexibility.  Each section is comprised of 4 subsections and an exercise set.  The first 3 subsections may be essential to later work, but the fourth subsection is not essential to later work and can either be covered briefly or even omitted.  In general, these subsections are devoted to additional graphical and numerical techniques, proofs of theorems, additional insights into previous material, and alternative techniques and identities. 

          Thus, it is conceivable that an instructor could progress through the course at breakneck speed by simply covering the first 3 subsections of each section.  Or more desirably, an instructor could choose what topics to emphasize and how much coverage to provide to each topic.  In either circumstance, the instructor can choose exercises and applications that best suit the needs of the students, whether they be mathematics majors, aspiring scientists, engineering students, or future businessmen.


Calculus as the Foundation of Modern Science and Math

          This textbook also presents a calculus course that best serves the needs of science and mathematics as it enters the twenty-first century.  To do so, we recognized that differential equations and integration were central to Calculus in its inception and have remained in the center ever since.  In this textbook, these two themes are often used to motivate both techniques and applications of calculus.  Much of the differential calculus is motivated by concepts related to differential equations, and once the fundamental theorem is introduced, much of the material is motivated by applications of the integral.

          Moreover, many of the topics covered in the later sections of the book are relatively new to calculus.  There are sections on mathematical modeling, discrete dynamical systems, Fourier series, and digital filtering.  The multivariable chapters include concepts like separation of variables to solve partial differential equations and the fundamental form of a surface.  In fact, the multivariable chapters constitute an online multivariable calculus course that is located at .

          We also realized that one of the major goals of any calculus course is that of preparing students for further study in mathematics, science, and engineering.  As a result, we have for several years worked with students to develop definitions of concepts that reflect modern treatments of those concepts while remaining accessible to the average student.   For example, open intervals are incorporated into the definition of the limit, thus giving it a slightly more topological flavor.  The definitions of differentiability and integrability are independent of the definitions of the derivative and the integral, which reflects more advanced treatments of differentiation and integration.  Indeed, the definite integral of a function is defined to be a limit of simple function approximations, thus preparing students for future work with modern definitions of the integral.

          Finally, the textbook also contains many applications of calculus that are currently relevant, including mathematical biology, mathematical modeling, geometric probability, curve-fitting, quantum mechanics, and a host of others.  There is also a capstone chapter after the multivariable chapters that applies all the calculus presented in the textbook to the analysis of the inverse square law and its many applications.


Implementation of the Plan

          Although the textbook is based on models of mathematical learning and the desire for relevant content, the original plan had to be modified due to pragmatic considerations.  For example, modifications were made to address the needs of AP calculus courses. In addition, the original organization of the book was altered so that it more closely resembled the content organization of other calculus textbooks.  In addition, we made a conscious effort to use the “rule of 3” whenever possible, which is to say that many concepts are presented numerically, graphically, and analytically.  In some sections, the use of technology is essential both to a presentation of the material and in the exercises, and in other sections, the use of technology is deprecated in favor of traditional pencil and paper skill development.

          In addition, we feel that the ability to read and write mathematics is essential in today’s world.  We have worked closely with students in developing the writing style for the text, and the result has been found to be very readable.  Also, throughout the text are various “Write to Learn” exercises that ask for students to write short essays communicating their understanding of a given problem or concept.  There are also short essays called “Next Steps” which are themselves followed by a collection of “Write to Learn” and group exercises. 

          There is extensive review material in the textbook.  Placement of precalculus review material corresponds roughly to its initial occurrence in the study of calculus.  We feel that this serves the students better than an all-at-once review at the beginning that is forgotten by the time those topics appear later on.

          Thus, this textbook augments our plan with what we feel to be the best from traditional textbooks, the reform movement, and the use of technology.  Moreover, it was designed to address both pedagogical and pragmatic considerations.  Finally, this textbook attempts to fully develop student comprehension, thus leading the student to appreciate that calculus remains a field of study that is growing and thriving, relevant and strong.

Structure of the Textbook

    The textbook is highly structured and the content of the course is rigorously organized.  Let’s begin our description of the structure of the textbook by examining the organization of individual sections.  In particular, each section is organized according to the following:

1.        Each Section has 4 subsections:  Each subsection introduces one or two new concepts followed by examples of these new ideas follow.

2.        The 4th Subsection typically presents ideas that are not necessary in later sections.  The fourth subsection often consists of proofs of theorems, additional applications, or additional examples, thus giving an instructor some discretion in how best to teach the course.

3.        After each of the 1st three subsections, there is a “Check your Reading” question.  These questions assess a student’s comprehension of the material just read and can be used to facilitate either discussion either in class or online via threaded message forums.

4.        Exercise Sets are Graded and Correspond Closely to the Examples:  These problem sets drill the techniques encountered in the section, whether they be graphical, numerical or analytical.

5.        Applications problems include “Write to Learn” and Discussion Problems:  In addition, we periodically include problems that are more challenging than usual. These are marked by an asterisk (*). 

The sections are written in what we call a “tutorial style.”  In particular, the sections have been designed to be as readable as possible, and the examples are written to be as self-explanatory as possible.

          The coverage of integration in this textbook also differs from traditional treatments, which in large part is due to our desire for this textbook to introduce calculus as a foundation for modern mathematics and science.  Although definite integrals are defined to be limits of Riemann sums and the fundamental theorem is proven rigorously, definite integral is defined to be a limit of simple function approximations.  While such a definition is no more difficult for students than the typical Cauchy-style definition with Riemann sums, it better prepares future mathematicians, statisticians, and engineers for the modern concepts of integration and function approximation they will encounter later.  It must be emphasized that we have been using this textbook for several semesters, and while our treatment of integration is different, students do not find it especially difficult.


Uses of Technology

          Technology is utilized throughout the textbook.  Indeed, the multivariable portions of the textbook, chapters 9 – 13, are being taught as a web-based, technology-intensive course, and will be available either in printed form or as on online course once the textbook is published.  In the single-variable portions of the book, chapters 1-8, technology is used in at least 3 different ways:

1.        Graphing Functions:  Graphing is used for verification, for exploration, and for problem solving. For example, graphing is used to develop and utilize the formal definition of the limit.

2.        Constructing Tables of Numerical Values:  In the business world, one often “runs the numbers to see what they say.”  We likewise see great value having students produce tables of numerical values when they are introduced to a concept. For example, in the multivariable sections we use tables of numerical values to explore limits in two variables.

3.        Symbolic Calculation:  When computer algebra systems are used to solve problems other than rote calculations, they can reinforce both a concept and its notation.  For example, we suggest the use of computer algebra systems for optimization problems in which the derivatives can be very difficult to compute by hand.   

However, although technology is utilized throughout the textbook, the single variable portions can be completed with no more than a simple graphing calculator (i.e., one without symbolic capabilities).



          Our textbook is not traditional, nor was it written as an action or reaction to any movement.  Instead, this book is an implementation of a plan developed through years of research.   We identified best practices in traditional approaches, the reform movement, and the use of instructional technology.  We collected information and developed models of how students learn mathematics.  We reviewed the use of calculus concepts in modern mathematics, science, and engineering.   

          We also tried to write the textbook that would best meet the needs of the wide variety of instructors who would be using it.  Along these lines, we would like to acknowledge the many contributions of those who explored and reviewed this project during its development.  Our treatment of the limit concept was greatly improved by insights and examples from A. Shadi Tahvildar-Zadeh.  Many improvements in the first two chapters are due to insights from Dr. Debra Knisley.  The use of technology was much improved by techniques and insights from Dr. Lyndell Kerley.   Many others have also contributed (and their names will be listed below once the book is published). 

          We thank all those who have contributed to this project, and we recommend it to anyone who wants to present calculus in an accessible, coherent, and relevant fashion.

[1] COPYRIGHT © 1999 by Jeff Knisley, East Tennessee State University.  All rights reserved.